http://primes.utm.edu/top20/page.php?id=2
"

Around 1825 Sophie Germain proved that the first case of Fermat's

Last Theorem is true for such primes. Soon after Legendre began to

generalize this by showing the first case of FLT also holds for odd

primes p such that kp+1 is prime, k=4, 8, 10, 14 and 16. In 1991 Fee

and Granville [FG91] extended this to k<100, k not a multiple of

three."

My question is: What about the converse? How many things does the

FLT now prove about primes? Not just about SG primes and the others

which were use to reduece the problem, I am thinking of what other

ways it may be used. Like: Is there any type of primes of a form

which includes x^n + y^n?